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This is a bit of a stretch, but bear with me. Elizabeth I's main choice between heirs in 1600 were as follows:
James VI, King of Scots (eldest male from Henry VIII's elder sister, Margaret, Queen of Scots' line from both her 1e and 2e marriages)
Arabella Stewart (eldest female, descended from a line junior to James', but only through Margaret's 2e marriage)
Edward Seymour, Baron Beauchamp (eldest male from Henry VIII's younger sister, Mary, Queen of France's line. However, his parents' marriage was invalid).
Anne Stanley, Baroness Chandos (eldest heir descended from a legitimate marriage from Mary, Queen of France)
Of these, most were Protestant. Yet Arabella and Baroness Chandos' father were both implicated in Catholic plans (Arabella was to be made queen of England and married to the duke of Parma's brother, for instance; Lord Stanley in the Heskeith Plot). James' mother was Catholic and his wife later converted, plus he seemed very interested in finding a Catholic match for his son(s). Seymour I can't find anything on (religiously speaking).
Now, say all four of these main heirs somehow end up going Catholic at some point during Elizabeth's reign. That it becomes clear to her ministers that the next person, with a claim to the throne of England, is a Catholic. What happens next?